Basic Euclidean Geometry
|
|
- Alexandrina Allison
- 5 years ago
- Views:
Transcription
1 hapter 1 asic Euclidean Geometry This chapter is not intended to be a complete survey of basic Euclidean Geometry, but rather a review for those who have previously taken a geometry course For a definitive account, see Euclid s Elements 11 Triangles triangle is a (plane) figure bounded by three line segments The most important result about triangles is that the sum of the angles of a triangle has measure equal to two right angles (or 180 ) This can be deduced from Fig 11, where the line DE is parallel to the line segment D E Fig 11 If one of the angles of a triangle is a right angle, we call the the triangle a right triangle For such triangles, we have Pythagoras s Theorem which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd 1
2 2 Explorations in Geometry Fig 12 In Fig 12, we have that = 2 12 Similar Triangles We start with two triangles and The definition of two triangles being similar can be given in one of two ways: We say that the triangles and are similar if either (1) the sides of the triangle are in proportion; that is, if = = ; or (2) the angles of the triangles are equal; that is, if =, =, and = It is easy to see that the two definitions are equivalent Thus, showing either relationship gives the other We ask the question: what is sufficient to show that two triangles are similar? Do we have to show that all three angles are equal; do we have to show that all three ratios are equal? In the case of angles, it is sufficient to show that two of the angles are equal, since this will automatically give that the third angles are equal (Why?) EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
3 asic Euclidean Geometry 3 However, it is not sufficient to show that two of the ratios of the sides are equal To see this, simply consider two isosceles triangles, and with = and = Then, =, but and will only be similar if = Problem 11 Let be a triangle Points and are chosen on and, respectively, such that = Prove that triangles and are similar 13 ongruent Triangles Similar triangles can be of different sizes (We see that in the diagram above) If the ratio of the sides has value 1, then we say that the triangles are congruent This means that the two triangles are identical in every way, although their orientation and position may differ There are several conditions that are sufficient for showing that two triangles are congruent They are 1 Three sides equal SSS 2 Two sides and the included angle SS 3 Two angles and the corresponding side S In the third case, we may as well assume that the side is common to both angles; hence, the notation S We do not insist on the same orientation for congruence EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
4 4 Explorations in Geometry 14 Quadrilaterals quadrilateral is a plane figure bounded by four line segments In this section we look briefly at some particular quadrilaterals (1) Trapezoid trapezoid is a quadrilateral with one pair of (opposite) sides parallel a symmetric trapezoid is a trapezoid where the non-parallel sides are equally inclined to the other sides See Fig 13 D Fig 13 In this example, we have both that = D and that D = D We also see that the sum of the opposite angle of a symmetric trapezoid is equal to two right angles (π) (2) Parallelogram parallelogram is a quadrilateral where the opposite sides are parallel This gives the result that opposite angle are equal See Fig 14 D Fig 14 EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
5 asic Euclidean Geometry 5 (3) Rhombus rhombus is a parallelogram where all sides are equal (4) Square square is a particular quadrilateral where all sides are equal and all angles are equal to one right angle (π/2) lternatively, a square is a rhombus where an angle is a right angle (which gives all angles as right angles) 15 Polygons polygon is a plane figure bounded by line segments Some special names are: Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon n Polygon basic result concerns the sum of the interior angles of a polygon The value, (2n 4) right angles, or (n 2)π, can be seen easily by triangulation: see Fig 15 Fig 15 EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
6 6 Explorations in Geometry Here we have n triangles The total measure of the angles is then n π From this, we must subtract the sum of the angles at the interior point, that is, 2π We shall be interested in a later chapter in regular polygons, and which can be constructed using straight edge and compasses 16 ircles and ngles In this section, we will recall a number of important results concerning angles, which arise naturally in the study of circles The first concerns the size of an angle in a semicircle Theorem 11 If is the diameter of a circle and is any point on the circle distinct from and, then = π/2 (in radian measure) Proof Let O be the centre of the circle to help, join O, and See Fig 16 O Fig 16 Since O = O, we have O = O Similarly, since O = O, we have O = O ut we know that O + O + O + O = π (sum of the interior angles of a triangle) EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
7 asic Euclidean Geometry 7 Hence, 2 O + 2 O = π, giving O + O = π/2 Therefore, = π/2 as desired Next, we consider the relationship between the angle subtended by an arc to a point on the circumference with the angle subtended by the same arc at the centre Theorem 12 Let be a chord of a circle (with centre O) which is not a diameter, and let be any point on the circle distinct from and (1) If is on the same side of as O, then O = 2 (2) If is on the opposite side of from O, then O = 2π 2 Proof Let be on the same side of as O To help, join O, O, and First, assume that does not intersect the radius O, and that does not intersect the radius O as illustrated in Fig 17 O Fig 17 Here, we have O = π ( O + O) We also have O + O + O + O = π ( O + O) Hence, O = O + O + O + O Since O = O, we have O = O, and since O = O, we have O = O Thus, O = 2( O + O) = 2 EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
8 8 Explorations in Geometry Next, assume that intersects the radius O as illustrated below (the case where intersects O is proved similarly, and is left as an exercise): see Fig 18 O Fig 18 O Fig 19 gain, we have O = π ( O + O) EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
9 asic Euclidean Geometry 9 ut this time, we have ( O O) + ( O O) + ( O + O) = π Thus, O = O + O O + O s before, we have O = O and O = O Hence, O = 2 O 2 O = 2 Now, let be on the opposite side of from the centre, and again, join O, O,, and O See Fig 19 Here, we have O = 2π ( O + O + O + O) Since O = O, we have O = O, and since O = O, we have O = O Thus, O = 2π 2 O 2 O = 2π 2 as desired The next result, which follows immediately from theorem 12, will often be more important in applications Theorem 13 If is a chord of a circle and and D are two distinct points on the circle, distinct from and, both of which lie on the same side of, then = D Proof If is a diameter, then the result follows from theorem 11, since both and D are right angles D O Fig 110 The ow-tie Lemma EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
10 10 Explorations in Geometry ssume, from now on, that is not a diameter Let O be the centre of the circle If and D are on the same side of as O, then theorem 12 (1) gives that 2 = O and 2 D = O, yielding that = D If and D are on the opposite side of from O, then theorem 12 (2) enables us also to conclude that = D If and D are on opposite sides of, we have the following: Theorem 14 If is a chord of a circle and and D are two points on the circle, distinct for and, lying on opposite sides of, then + D = π Proof s in the previous result, if is a diameter, then + D = π/2 + π/2 = π ssume for the remainder of this proof that is not a diameter, and let O be the centre of the circle Without loss of generality, assume that and O lie on the same side of Then, theorem 12 tells us that O = 2 and that O = 2π 2 D Hence, 2 = 2π 2 D, yielding that + D = π Finally, we will make an interesting observation involving the angles between chords and tangents Theorem 15 Let, and be any three points on a circle, and let D be such that D is tangent to the circle (at ) and that D is on the opposite side of line from Then D = Proof First note that if is a diameter of the circle, then = π/2 by theorem 11, and D = π/2 since D is a tangent This, the result holds Henceforth, we assume that is not a diameter Draw the diameter at and call it X We consider two cases either X intersects the chord or it does not First, assume that X intersects Draw the chord X See Fig 110 Note that X and DX are both right angles lso, by theorem 13, we have = X Hence, D = π/2 X = X =, as desired Next, assume that X does not intersect, and again, draw the chord X See Fig 111 s above, X and DX are right angles EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
11 asic Euclidean Geometry 11 D X Fig 111 In this case, + X = π by theorem 14 Hence, D = π/2 + X = π/2 + (π/2 X) = π (π ) = 17 yclic Quadrilaterals quadrilateral whose vertices all lie on a circle is called a cyclic quadrilateral Now, any triangle has the property that a unique circle can be drawn through its vertices (for the demonstration, theorem 36 on page 55) Thus, to say that a quadrilateral is cyclic is really quite restrictive in fact, we are demanding that D lie on the unique circle passing through, and Similarly, we could start with the unique circle through any other three of the four named points Therefore, we should expect that cyclic quadrilaterals have some very particular properties One such property is clear from the geometry of a circle, we see that each angle of a cyclic quadrilateral D must be less than π See Fig 112 To facilitate terminology, we call any quadrilateral with this property convex in fact, there is a more general definition of convex which applies EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
12 12 Explorations in Geometry X D Fig 112 to all polygons, but in the case of quadrilaterals, it is equivalent to ours Two other important properties of cyclic quadrilaterals follow immediately from other result proved earlier In fact, theorems 13 and 14 can be re-stated as follows (we assume the vertices to be labelled in a clockwise order): (1) If D is a cyclic quadrilateral, then = D (2) If D is a cyclic quadrilateral, then + D = π Note that the first of these results also has three other possible conclusions, depending on which of, D or D is chosen as the chord in theorem 13 For example, D = D s an exercise, the reader should list all possibilities Similarly to the second result, we also have D + D = π ut this could also be deduced from the fact that the sum of the interior angles of a quadrilateral is 2π It is important for subsequent material to note that the converses of these two results hold Theorem 16 Suppose that D is a convex quadrilateral such that = D Then quadrilateral D is a cyclic quadrilateral EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
13 asic Euclidean Geometry 13 D Fig 113 Proof Draw the unique circle through, and We wish to prove that D lies on the circle See Fig 113 Let E be the second point of intersection of the circle with D ( being the first) and join E Since D is convex, the point E does exist and is on the same side of as is D It follows that if D E, then E D ut, D is a cyclic quadrilateral, and thus, we have E = Since = D (as marked), this is a contradiction Hence, D = E Theorem 17 If D is a quadrilateral in which + D = π, then D is cyclic Proof Note that the given condition forces quadrilateral D to be convex See Fig 114 s with theorem 16, draw the unique circle through, and, and let E be the second point of intersection of the circle with D Join E and If D E, then E D ut E is cyclic, yielding that + E = π This implies that E = D, which is a contradiction EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
14 14 Explorations in Geometry E D Fig Intersecting hords We consider a circle with two intersecting chords Suppose that and D are two chords of a given circle which intersect at some point X inside the circle See Fig 115 Here, we observe that theorem 13 implies that D = D, and thus, it follows that triangles X and DX are similar Hence, we have that X, or, equivalently, XD = X X X X = X XD This result is known as the Intersecting hords Theorem Of course, it can be immediately interpreted as a result about the diagonals of any cyclic quadrilateral Problem 12 In equilateral triangle of side length 2, suppose that M and N are the mid-points of and, respectively The triangle is inscribed in a circle The line segment MN is extended to meet the circle at P Determine the length of the line segment NP (Solution on page 157) Problem 13 Prove the converse of the Intersecting hords Theorem That is, prove that if D is a convex quadrilateral, if X is the point of EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
15 asic Euclidean Geometry 15 E D Fig 115 X D Fig 116 intersection of the diagonals and D, and if X X = DX X, then D is a cyclic quadrilateral EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
16 16 Explorations in Geometry 19 Inversion The use of inversion can be very useful in solving some problems We give the basic ideas here We work in the Euclidean plane, with one ideal point at infinity Roughly speaking, an inversion is a transformation of the plane that generalizes the idea of reflection in a line 191 Reflection in a line In reflection in a line l, a point X is mapped to a point X that is the same distance from the line l as is X, but is in the opposite half plane to X l X X Fig Inversion in a circle Generalize this notion of reflection by replacing the line l by a circle Γ m α β X X l Fig 118 Suppose that m XX, meeting l at P Reflecting P X X gives P XX, which are equal Since m XX, we have α = β, so that P XX = α = β Now, let your imagination expand so that l is an infinitely large circle, with m lying on the radius of this circle through the point P We now view is thus: EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
17 asic Euclidean Geometry 17 P m r O X X Fig 119 We have P OX similar to XOP, giving OX OP = OP OX, so that OX OX = OP 2 = r 2 Now, suppose that O is a fixed point (called the centre of inversion) and that c is a fixed positive number (called the radius of inversion) Definition 11 P and Q are inverses with respect to O with radius c if OPOQ = c 2 Definition 12 The circle centre O and radius c is called the circle of inversion Theorem 18 The circle of inversion is invariant under inversion Theorem 19 The inverse of a line through the centre of inversion is that line (ut it is not an invariant) The proofs of these theorems are left to the reader Theorem 110 The inverse of a line, not through the centre of inversion, is a circle passing through the centre of inversion Proof Let O be the centre of inversion and P Q be the line, such that OP P Q Let Γ be the circle of inversion Let P and Q be the inverses of P and Q, respectively Then OPOP = OQOQ, giving OP OQ = OP OQ Thus, triangles OP Q and OQ P are similar Since QP O = 90, we have that P Q O = 90, giving that O, P and Q lie on a circle of diameter OP The converse is also true EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
18 18 Explorations in Geometry Q Q P P O Fig 120 Theorem 110 Γ Theorem 111 The inverse of a circle not passing through the centre of inversion in a circle (not passing through the centre of inversion) Proof Γ O Q P Q P Fig 121 Theorem 111 Let O be the centre of inversion and let the given circle have centre Then OPOP = OQOQ = c 2 lso, note that OPOQ = O 2 = k 2, where O is the tangent from O to the given circle Thus, c 4 = (OPOP ) (OQOQ ) = (OP OQ ) (OPOQ) = (OP OQ ) k 2, whence, OP OQ = k2 c 4 = (O ) 2 This means that the image is a circle! EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
19 asic Euclidean Geometry 19 Now, here is a result for you to try to obtain yourself! Problem 14 The measure of the angle between two intersecting circles is invariant under inversion Here is a useful figure Problem 15 Two points, and are distinct and not collinear with the centre O of the circle of inversion The images of the two points are and, respectively Prove that triangles O and O are similar Note that their orientations are reversed Problem 16 straight line passing through the centre of the circle of inversion maps onto itself EXPLORTIONS IN GEOMETRY World Scientific Publishing o Pte Ltd
SOME IMPORTANT PROPERTIES/CONCEPTS OF GEOMETRY (Compiled by Ronnie Bansal)
1 SOME IMPORTANT PROPERTIES/CONCEPTS OF GEOMETRY (Compiled by Ronnie Bansal) 1. Basic Terms and Definitions: a) Line-segment: A part of a line with two end points is called a line-segment. b) Ray: A part
More informationThe angle measure at for example the vertex A is denoted by m A, or m BAC.
MT 200 ourse notes on Geometry 5 2. Triangles and congruence of triangles 2.1. asic measurements. Three distinct lines, a, b and c, no two of which are parallel, form a triangle. That is, they divide the
More informationAngles. An angle is: the union of two rays having a common vertex.
Angles An angle is: the union of two rays having a common vertex. Angles can be measured in both degrees and radians. A circle of 360 in radian measure is equal to 2π radians. If you draw a circle with
More informationInversive Plane Geometry
Inversive Plane Geometry An inversive plane is a geometry with three undefined notions: points, circles, and an incidence relation between points and circles, satisfying the following three axioms: (I.1)
More informationMATH 30 GEOMETRY UNIT OUTLINE AND DEFINITIONS Prepared by: Mr. F.
1 MTH 30 GEMETRY UNIT UTLINE ND DEFINITINS Prepared by: Mr. F. Some f The Typical Geometric Properties We Will Investigate: The converse holds in many cases too! The Measure f The entral ngle Tangent To
More informationMANHATTAN HUNTER SCIENCE HIGH SCHOOL GEOMETRY CURRICULUM
COORDINATE Geometry Plotting points on the coordinate plane. Using the Distance Formula: Investigate, and apply the Pythagorean Theorem as it relates to the distance formula. (G.GPE.7, 8.G.B.7, 8.G.B.8)
More informationElementary Planar Geometry
Elementary Planar Geometry What is a geometric solid? It is the part of space occupied by a physical object. A geometric solid is separated from the surrounding space by a surface. A part of the surface
More informationCURRICULUM GUIDE. Honors Geometry
CURRICULUM GUIDE Honors Geometry This level of Geometry is approached at an accelerated pace. Topics of postulates, theorems and proofs are discussed both traditionally and with a discovery approach. The
More informationInstructional Unit CPM Geometry Unit Content Objective Performance Indicator Performance Task State Standards Code:
306 Instructional Unit Area 1. Areas of Squares and The students will be -Find the amount of carpet 2.4.11 E Rectangles able to determine the needed to cover various plane 2. Areas of Parallelograms and
More informationMadison County Schools Suggested Geometry Pacing Guide,
Madison County Schools Suggested Geometry Pacing Guide, 2016 2017 Domain Abbreviation Congruence G-CO Similarity, Right Triangles, and Trigonometry G-SRT Modeling with Geometry *G-MG Geometric Measurement
More informationadded to equal quantities, their sum is equal. Same holds for congruence.
Mr. Cheung s Geometry Cheat Sheet Theorem List Version 6.0 Updated 3/14/14 (The following is to be used as a guideline. The rest you need to look up on your own, but hopefully this will help. The original
More informationGeometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review
Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Polygon a closed plane figure with at least 3 sides that are segments -the sides do not intersect except at the vertices N-gon -
More informationLines Plane A flat surface that has no thickness and extends forever.
Lines Plane A flat surface that has no thickness and extends forever. Point an exact location Line a straight path that has no thickness and extends forever in opposite directions Ray Part of a line that
More informationGEOMETRY is the study of points in space
CHAPTER 5 Logic and Geometry SECTION 5-1 Elements of Geometry GEOMETRY is the study of points in space POINT indicates a specific location and is represented by a dot and a letter R S T LINE is a set of
More informationPostulates, Theorems, and Corollaries. Chapter 1
Chapter 1 Post. 1-1-1 Through any two points there is exactly one line. Post. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. Post. 1-1-3 If two points lie in a
More informationGeometry: Traditional Pathway
GEOMETRY: CONGRUENCE G.CO Prove geometric theorems. Focus on validity of underlying reasoning while using variety of ways of writing proofs. G.CO.11 Prove theorems about parallelograms. Theorems include:
More informationTopics in geometry Exam 1 Solutions 7/8/4
Topics in geometry Exam 1 Solutions 7/8/4 Question 1 Consider the following axioms for a geometry: There are exactly five points. There are exactly five lines. Each point lies on exactly three lines. Each
More informationSOL Chapter Due Date
Name: Block: Date: Geometry SOL Review SOL Chapter Due Date G.1 2.2-2.4 G.2 3.1-3.5 G.3 1.3, 4.8, 6.7, 9 G.4 N/A G.5 5.5 G.6 4.1-4.7 G.7 6.1-6.6 G.8 7.1-7.7 G.9 8.2-8.6 G.10 1.6, 8.1 G.11 10.1-10.6, 11.5,
More information2.1 Angles, Lines and Parallels & 2.2 Congruent Triangles and Pasch s Axiom
2 Euclidean Geometry In the previous section we gave a sketch overview of the early parts of Euclid s Elements. While the Elements set the standard for the modern axiomatic approach to mathematics, it
More informationGeometry. Geometry. Domain Cluster Standard. Congruence (G CO)
Domain Cluster Standard 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance
More informationGEOMETRY CURRICULUM MAP
2017-2018 MATHEMATICS GEOMETRY CURRICULUM MAP Department of Curriculum and Instruction RCCSD Congruence Understand congruence in terms of rigid motions Prove geometric theorems Common Core Major Emphasis
More informationEUCLID S GEOMETRY. Raymond Hoobler. January 27, 2008
EUCLID S GEOMETRY Raymond Hoobler January 27, 2008 Euclid rst codi ed the procedures and results of geometry, and he did such a good job that even today it is hard to improve on his presentation. He lived
More informationPearson Mathematics Geometry Common Core 2015
A Correlation of Pearson Mathematics Geometry Common Core 2015 to the Common Core State Standards for Bid Category 13-040-10 A Correlation of Pearson, Common Core Pearson Geometry Congruence G-CO Experiment
More informationPi at School. Arindama Singh Department of Mathematics Indian Institute of Technology Madras Chennai , India
Pi at School rindama Singh epartment of Mathematics Indian Institute of Technology Madras Chennai-600036, India Email: asingh@iitm.ac.in bstract: In this paper, an attempt has been made to define π by
More informationChapter 1. acute angle (A), (G) An angle whose measure is greater than 0 and less than 90.
hapter 1 acute angle (), (G) n angle whose measure is greater than 0 and less than 90. adjacent angles (), (G), (2T) Two coplanar angles that share a common vertex and a common side but have no common
More informationVideos, Constructions, Definitions, Postulates, Theorems, and Properties
Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos Proof Overview: http://tinyurl.com/riehlproof Modules 9 and 10: http://tinyurl.com/riehlproof2 Module 9 Review: http://tinyurl.com/module9livelesson-recording
More informationStandards to Topics. Common Core State Standards 2010 Geometry
Standards to Topics G-CO.01 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance
More informationPlane Geometry. Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2011
lane Geometry aul Yiu epartment of Mathematics Florida tlantic University Summer 2011 NTENTS 101 Theorem 1 If a straight line stands on another straight line, the sum of the adjacent angles so formed is
More informationCONSTRUCTIONS Introduction Division of a Line Segment
216 MATHEMATICS CONSTRUCTIONS 11 111 Introduction In Class IX, you have done certain constructions using a straight edge (ruler) and a compass, eg, bisecting an angle, drawing the perpendicular bisector
More informationCarnegie Learning High School Math Series: Geometry Indiana Standards Worktext Correlations
Carnegie Learning High School Math Series: Logic and Proofs G.LP.1 Understand and describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates,
More informationSupporting planning for shape, space and measures in Key Stage 4: objectives and key indicators
1 of 7 Supporting planning for shape, space and measures in Key Stage 4: objectives and key indicators This document provides objectives to support planning for shape, space and measures in Key Stage 4.
More informationGeometry GEOMETRY. Congruence
Geometry Geometry builds on Algebra I concepts and increases students knowledge of shapes and their properties through geometry-based applications, many of which are observable in aspects of everyday life.
More informationThomas Jefferson High School for Science and Technology Program of Studies TJ Math 1
Course Description: This course is designed for students who have successfully completed the standards for Honors Algebra I. Students will study geometric topics in depth, with a focus on building critical
More informationCommon Core Specifications for Geometry
1 Common Core Specifications for Geometry Examples of how to read the red references: Congruence (G-Co) 2-03 indicates this spec is implemented in Unit 3, Lesson 2. IDT_C indicates that this spec is implemented
More informationSuggested List of Mathematical Language. Geometry
Suggested List of Mathematical Language Geometry Problem Solving A additive property of equality algorithm apply constraints construct discover explore generalization inductive reasoning parameters reason
More informationIndex COPYRIGHTED MATERIAL. Symbols & Numerics
Symbols & Numerics. (dot) character, point representation, 37 symbol, perpendicular lines, 54 // (double forward slash) symbol, parallel lines, 54, 60 : (colon) character, ratio of quantity representation
More information7. The Gauss-Bonnet theorem
7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
More informationCourse: Geometry Level: Regular Date: 11/2016. Unit 1: Foundations for Geometry 13 Days 7 Days. Unit 2: Geometric Reasoning 15 Days 8 Days
Geometry Curriculum Chambersburg Area School District Course Map Timeline 2016 Units *Note: unit numbers are for reference only and do not indicate the order in which concepts need to be taught Suggested
More informationHonors Geometry Pacing Guide Honors Geometry Pacing First Nine Weeks
Unit Topic To recognize points, lines and planes. To be able to recognize and measure segments and angles. To classify angles and name the parts of a degree To recognize collinearity and betweenness of
More informationGeometry Vocabulary Math Fundamentals Reference Sheet Page 1
Math Fundamentals Reference Sheet Page 1 Acute Angle An angle whose measure is between 0 and 90 Acute Triangle A that has all acute Adjacent Alternate Interior Angle Two coplanar with a common vertex and
More informationChapter 10 Similarity
Chapter 10 Similarity Def: The ratio of the number a to the number b is the number. A proportion is an equality between ratios. a, b, c, and d are called the first, second, third, and fourth terms. The
More informationTest #1: Chapters 1, 2, 3 Test #2: Chapters 4, 7, 9 Test #3: Chapters 5, 6, 8 Test #4: Chapters 10, 11, 12
Progress Assessments When the standards in each grouping are taught completely the students should take the assessment. Each assessment should be given within 3 days of completing the assigned chapters.
More informationGrade IX. Mathematics Geometry Notes. #GrowWithGreen
Grade IX Mathematics Geometry Notes #GrowWithGreen The distance of a point from the y - axis is called its x -coordinate, or abscissa, and the distance of the point from the x -axis is called its y-coordinate,
More informationPearson Mathematics Geometry
A Correlation of Pearson Mathematics Geometry Indiana 2017 To the INDIANA ACADEMIC STANDARDS Mathematics (2014) Geometry The following shows where all of the standards that are part of the Indiana Mathematics
More informationGeometry. Cluster: Experiment with transformations in the plane. G.CO.1 G.CO.2. Common Core Institute
Geometry Cluster: Experiment with transformations in the plane. G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of
More informationGeometry Basics * Rory Adams Free High School Science Texts Project Mark Horner Heather Williams. 1 Introduction. 2 Points and Lines
OpenStax-NX module: m31494 1 Geometry asics * Rory dams Free High School Science Texts Project Mark Horner Heather Williams This work is produced by OpenStax-NX and licensed under the reative ommons ttribution
More informationCourse: Geometry PAP Prosper ISD Course Map Grade Level: Estimated Time Frame 6-7 Block Days. Unit Title
Unit Title Unit 1: Geometric Structure Estimated Time Frame 6-7 Block 1 st 9 weeks Description of What Students will Focus on on the terms and statements that are the basis for geometry. able to use terms
More informationGeometry/Pre AP Geometry Common Core Standards
1st Nine Weeks Transformations Transformations *Rotations *Dilation (of figures and lines) *Translation *Flip G.CO.1 Experiment with transformations in the plane. Know precise definitions of angle, circle,
More informationMATH 113 Section 8.2: Two-Dimensional Figures
MATH 113 Section 8.2: Two-Dimensional Figures Prof. Jonathan Duncan Walla Walla University Winter Quarter, 2008 Outline 1 Classifying Two-Dimensional Shapes 2 Polygons Triangles Quadrilaterals 3 Other
More informationWAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ACCELERATED GEOMETRY (June 2014)
UNIT: Chapter 1 Essentials of Geometry UNIT : How do we describe and measure geometric figures? Identify Points, Lines, and Planes (1.1) How do you name geometric figures? Undefined Terms Point Line Plane
More informationGeometry. Geometry is one of the most important topics of Quantitative Aptitude section.
Geometry Geometry is one of the most important topics of Quantitative Aptitude section. Lines and Angles Sum of the angles in a straight line is 180 Vertically opposite angles are always equal. If any
More informationMADISON ACADEMY GEOMETRY PACING GUIDE
MADISON ACADEMY GEOMETRY PACING GUIDE 2018-2019 Standards (ACT included) ALCOS#1 Know the precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined
More informationGEOMETRY Curriculum Overview
GEOMETRY Curriculum Overview Semester 1 Semester 2 Unit 1 ( 5 1/2 Weeks) Unit 2 Unit 3 (2 Weeks) Unit 4 (1 1/2 Weeks) Unit 5 (Semester Break Divides Unit) Unit 6 ( 2 Weeks) Unit 7 (7 Weeks) Lines and Angles,
More informationUnit Activity Correlations to Common Core State Standards. Geometry. Table of Contents. Geometry 1 Statistics and Probability 8
Unit Activity Correlations to Common Core State Standards Geometry Table of Contents Geometry 1 Statistics and Probability 8 Geometry Experiment with transformations in the plane 1. Know precise definitions
More informationCourse Number: Course Title: Geometry
Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line
More informationMathematics High School Geometry An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts
Mathematics High School Geometry An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts interpreting a schematic drawing, estimating the amount of
More informationPASS. 5.2.b Use transformations (reflection, rotation, translation) on geometric figures to solve problems within coordinate geometry.
Geometry Name Oklahoma cademic tandards for Oklahoma P PRCC odel Content Frameworks Current ajor Curriculum Topics G.CO.01 Experiment with transformations in the plane. Know precise definitions of angle,
More informationCCSD Proficiency Scale - Language of Geometry
CCSD Scale - Language of Geometry Content Area: HS Math Grade : Geometry Standard Code: G-CO.1 application G-CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line
More informationRectilinear Figures. Introduction
2 Rectilinear Figures Introduction If we put the sharp tip of a pencil on a sheet of paper and move from one point to the other, without lifting the pencil, then the shapes so formed are called plane curves.
More informationPrentice Hall Mathematics Geometry, Foundations Series 2011
Prentice Hall Mathematics Geometry, Foundations Series 2011 Geometry C O R R E L A T E D T O from March 2009 Geometry G.1 Points, Lines, Angles and Planes G.1.1 Find the length of line segments in one-
More informationHigh School Mathematics Geometry Vocabulary Word Wall Cards
High School Mathematics Geometry Vocabulary Word Wall Cards Table of Contents Reasoning, Lines, and Transformations Basics of Geometry 1 Basics of Geometry 2 Geometry Notation Logic Notation Set Notation
More informationNEW YORK GEOMETRY TABLE OF CONTENTS
NEW YORK GEOMETRY TABLE OF CONTENTS CHAPTER 1 POINTS, LINES, & PLANES {G.G.21, G.G.27} TOPIC A: Concepts Relating to Points, Lines, and Planes PART 1: Basic Concepts and Definitions...1 PART 2: Concepts
More informationChapter 6. Sir Migo Mendoza
Circles Chapter 6 Sir Migo Mendoza Central Angles Lesson 6.1 Sir Migo Mendoza Central Angles Definition 5.1 Arc An arc is a part of a circle. Types of Arc Minor Arc Major Arc Semicircle Definition 5.2
More informationGeometry. (F) analyze mathematical relationships to connect and communicate mathematical ideas; and
(1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is (A) apply mathematics to problems arising in everyday life,
More informationGeometry Curriculum Map
Geometry Curriculum Map Unit 1 st Quarter Content/Vocabulary Assessment AZ Standards Addressed Essentials of Geometry 1. What are points, lines, and planes? 1. Identify Points, Lines, and Planes 1. Observation
More information8. T(3, 4) and W(2, 7) 9. C(5, 10) and D(6, -1)
Name: Period: Chapter 1: Essentials of Geometry In exercises 6-7, find the midpoint between the two points. 6. T(3, 9) and W(15, 5) 7. C(1, 4) and D(3, 2) In exercises 8-9, find the distance between the
More informationPrentice Hall CME Project Geometry 2009
Prentice Hall CME Project Geometry 2009 Geometry C O R R E L A T E D T O from March 2009 Geometry G.1 Points, Lines, Angles and Planes G.1.1 Find the length of line segments in one- or two-dimensional
More informationAnswers. (1) Parallelogram. Remember: A four-sided flat shape where the opposite sides are parallel is called a parallelogram. Here, AB DC and BC AD.
Answers (1) Parallelogram Remember: A four-sided flat shape where the opposite sides are parallel is called a parallelogram. Here, AB DC and BC AD. (2) straight angle The angle whose measure is 180 will
More informationMathematics Standards for High School Geometry
Mathematics Standards for High School Geometry Geometry is a course required for graduation and course is aligned with the College and Career Ready Standards for Mathematics in High School. Throughout
More informationGeometry: A Complete Course
Geometry: omplete ourse with Trigonometry) Module Progress Tests Written by: Larry. ollins Geometry: omplete ourse with Trigonometry) Module - Progress Tests opyright 2014 by VideotextInteractive Send
More informationGeometry: A Complete Course
Geometry: Complete Course with Trigonometry) Module E - Course Notes Written by: Thomas E. Clark Geometry: Complete Course with Trigonometry) Module E - Course Notes Copyright 2014 by VideotextInteractive
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for
More informationLesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms
Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms Getting Ready: How will you know whether or not a figure is a parallelogram? By definition, a quadrilateral is a parallelogram if it has
More informationUnit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with
Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with endpoints on the circle. Diameter - A chord which passes through
More informationMAT104: Fundamentals of Mathematics II Introductory Geometry Terminology Summary. Section 11-1: Basic Notions
MAT104: Fundamentals of Mathematics II Introductory Geometry Terminology Summary Section 11-1: Basic Notions Undefined Terms: Point; Line; Plane Collinear Points: points that lie on the same line Between[-ness]:
More informationMathematics High School Geometry
Mathematics High School Geometry An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts interpreting a schematic drawing, estimating the amount of
More informationHistory of Mathematics
History of Mathematics Paul Yiu Department of Mathematics Florida tlantic University Spring 2014 1: Pythagoras Theorem in Euclid s Elements Euclid s Elements n ancient Greek mathematical classic compiled
More informationTheorems & Postulates Math Fundamentals Reference Sheet Page 1
Math Fundamentals Reference Sheet Page 1 30-60 -90 Triangle In a 30-60 -90 triangle, the length of the hypotenuse is two times the length of the shorter leg, and the length of the longer leg is the length
More informationThe Question papers will be structured according to the weighting shown in the table below.
3. Time and Mark allocation The Question papers will be structured according to the weighting shown in the table below. DESCRIPTION Question Paper 1: Grade 12: Book work, e.g. proofs of formulae (Maximum
More informationAppendix E. Plane Geometry
Appendix E Plane Geometry A. Circle A circle is defined as a closed plane curve every point of which is equidistant from a fixed point within the curve. Figure E-1. Circle components. 1. Pi In mathematics,
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationPROPERTIES OF TRIANGLES AND QUADRILATERALS
Mathematics Revision Guides Properties of Triangles, Quadrilaterals and Polygons Page 1 of 22 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier PROPERTIES OF TRIANGLES AND QUADRILATERALS
More informationSelect the best answer. Bubble the corresponding choice on your scantron. Team 13. Geometry
Team Geometry . What is the sum of the interior angles of an equilateral triangle? a. 60 b. 90 c. 80 d. 60. The sine of angle A is. What is the cosine of angle A? 6 4 6 a. b. c.. A parallelogram has all
More informationCIRCLE. Circle is a collection of all points in a plane which are equidistant from a fixed point.
CIRCLE Circle is a collection of all points in a plane which are equidistant from a fixed point. The fixed point is called as the centre and the constant distance is called as the radius. Parts of a Circle
More informationCHAPTER 2 REVIEW COORDINATE GEOMETRY MATH Warm-Up: See Solved Homework questions. 2.2 Cartesian coordinate system
CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH6 2.1 Warm-Up: See Solved Homework questions 2.2 Cartesian coordinate system Coordinate axes: Two perpendicular lines that intersect at the origin O on each line.
More informationA VERTICAL LOOK AT KEY CONCEPTS AND PROCEDURES GEOMETRY
A VERTICAL LOOK AT KEY CONCEPTS AND PROCEDURES GEOMETRY Revised TEKS (2012): Building to Geometry Coordinate and Transformational Geometry A Vertical Look at Key Concepts and Procedures Derive and use
More informationGeometry Common Core State Standard (CCSS) Math
= ntroduced R=Reinforced/Reviewed HGH SCHOOL GEOMETRY MATH STANDARDS 1 2 3 4 Congruence Experiment with transformations in the plane G.CO.1 Know precise definitions of angle, circle, perpendicular line,
More informationCHAPTER 2. Euclidean Geometry
HPTER 2 Euclidean Geometry In this chapter we start off with a very brief review of basic properties of angles, lines, and parallels. When presenting such material, one has to make a choice. One can present
More information104, 107, 108, 109, 114, 119, , 129, 139, 141, , , , , 180, , , 128 Ch Ch1-36
111.41. Geometry, Adopted 2012 (One Credit). (c) Knowledge and skills. Student Text Practice Book Teacher Resource: Activities and Projects (1) Mathematical process standards. The student uses mathematical
More informationHoughton Mifflin Harcourt Geometry 2015 correlated to the New York Common Core Learning Standards for Mathematics Geometry
Houghton Mifflin Harcourt Geometry 2015 correlated to the New York Common Core Learning Standards for Mathematics Geometry Standards for Mathematical Practice SMP.1 Make sense of problems and persevere
More informationPolygons are named by the number of sides they have:
Unit 5 Lesson 1 Polygons and Angle Measures I. What is a polygon? (Page 322) A polygon is a figure that meets the following conditions: It is formed by or more segments called, such that no two sides with
More informationMCPS Geometry Pacing Guide Jennifer Mcghee
Units to be covered 1 st Semester: Units to be covered 2 nd Semester: Tools of Geometry; Logic; Constructions; Parallel and Perpendicular Lines; Relationships within Triangles; Similarity of Triangles
More informationGeometry Definitions and Theorems. Chapter 9. Definitions and Important Terms & Facts
Geometry Definitions and Theorems Chapter 9 Definitions and Important Terms & Facts A circle is the set of points in a plane at a given distance from a given point in that plane. The given point is the
More informationGEOMETRY CCR MATH STANDARDS
CONGRUENCE, PROOF, AND CONSTRUCTIONS M.GHS. M.GHS. M.GHS. GEOMETRY CCR MATH STANDARDS Mathematical Habits of Mind. Make sense of problems and persevere in solving them.. Use appropriate tools strategically..
More informationKillingly Public Schools. Grades Draft Sept. 2002
Killingly Public Schools Grades 10-12 Draft Sept. 2002 ESSENTIALS OF GEOMETRY Grades 10-12 Language of Plane Geometry CONTENT STANDARD 10-12 EG 1: The student will use the properties of points, lines,
More informationLesson 13.1 The Premises of Geometry
Lesson 13.1 The remises of Geometry 1. rovide the missing property of equality or arithmetic as a reason for each step to solve the equation. Solve for x: 5(x 4) 2x 17 Solution: 5(x 4) 2x 17 a. 5x 20 2x
More informationProblem 2.1. Complete the following proof of Euclid III.20, referring to the figure on page 1.
Math 3181 Dr. Franz Rothe October 30, 2015 All3181\3181_fall15t2.tex 2 Solution of Test Name: Figure 1: Central and circumference angle of a circular arc, both obtained as differences Problem 2.1. Complete
More informationUse throughout the course: for example, Parallel and Perpendicular Lines Proving Lines Parallel. Polygons and Parallelograms Parallelograms
Geometry Correlated to the Texas Essential Knowledge and Skills TEKS Units Lessons G.1 Mathematical Process Standards The student uses mathematical processes to acquire and demonstrate mathematical understanding.
More informationGeometry Geometry Grade Grade Grade
Grade Grade Grade 6.G.1 Find the area of right triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the
More informationTerm Definition Figure
Notes LT 1.1 - Distinguish and apply basic terms of geometry (coplanar, collinear, bisectors, congruency, parallel, perpendicular, etc.) Term Definition Figure collinear on the same line (note: you do
More information